1 Solution to Homework 4

Size: px
Start display at page:

Download "1 Solution to Homework 4"

Transcription

1 Solution to Homework Section. 5. The characteristic equation is r r + = (r )(r ) = 0 r = or r =. y(t) = c e t + c e t y = c e t + c e t. y(0) =, y (0) = c + c =, c + c = c =, c =. To find the maximum value of y(t), let To find the point where y(t) = 0, let y(t) = e t + e t. y (t) = e t + e t = 0 t = ln y M = y(t) = 9. y(t) = e t + e t = 0 t = ln. 6. The characteristic equation is r r = (r )(r + ) = 0 r = or r =. y(t) = c e t + c e t y = c e t c e t. y(0) = α, y (0) = c + c = α, c c = c = + α y(t) = + α e t + α e t. y(t) 0 as t if and only if the coefficient of e t in y(t) is 0, which requires, c = α. + α = 0 α =.. Section. x(x + ) + (x + )x = 0. x e x (x + ) x(x + )e x (x + ) + (x + )xe x = 0. Since y /y = e x is not a constant for x > 0, y and y are linearly independent, and so constitute a fundamental set of solutions.

2 5. 7. y x α(α + ) x y + x y = 0. p(x) = x x W = exp( p(x)dx) = e ln( x )+C = A x. y + t y + e t y = 0. p(t) = t W = exp( p(t)) = exp( t ) = e ln t+c = A t. W (y, y )() = A = A = W (y, y )(5) = A 5 = 5. Section. 7. The characteristic equation is r + r + = (r + ) + = 0 r = ± i. y(t) = c e t cos t + c e t sin t.. The characteristic equation is r r + 5 = (r ) + = 0 r = ± i. y(t) = c e t cos t + c e t sin t. y( π ) = 0 c = 0 y(t) = c e t sin t y (t) = c e t (sin t + cos t). y ( π ) = c = e π y(t) = e t π sin t. 7. Assume y(t) = t n is a solution to the equation, then n(n )t n + nt n + t n = t n (n + n + ) = t n (n + )(n + ) = 0 n = or n =. y(t) = c t + c t. Section. 7. The characteristic equation is 6r + r + 9 = (r + ) = 0 r = r =. y(t) = e t (c + c t).

3 . The characteristic equation is r r + = (r ) = 0 r = r =. y(t) = e t (c + c t) y (t) = e t (c + c + c t). y(0) =, y (0) = b c =, c + c = b c =, c = b. y(t) = e t [ + (b )t]. The critical b that separates y(t) between growing up and down as t satisfies b = 0 b =.. y x y + x y = 0. y = sin(x ), p(x) = x, W (x) = exp( W (x) y (x) = y (x) y(x) dx = sin(x ) p(x)dx) = x. csc (x )xdx = sin(x ) cot(x ) = cos(x ).. Assume y(t) = t n is a solution to the equation, then n(n )t n nt n + t n = t n (n n + ) = t n (n ) = 0 n = n =. y(t) = c t + c t ln t.

4 Solution to Homework 5 Section.5 5. First solve the homogeneous equation. y + y = 0 r(r + ) = 0 y = c + c e t. Next we find the particular solution to y + y = and y + y = sin t and add them up. For y + y =, since 0 is a single root of the characteristic equation, let Y = at. + Y = a = a = Y = t. For y +y = sin t, since i is not a root of the characteristic equation, let Y = a sin t+b cos t. + Y = a sin t b cos t + a cos t b sin t = sin t a = b =. y = c + c e t + Y + Y = c + c e t + 0. First solve the homogeneous equation. y + y + y = 0 (r + ) + 5 = 0 r = ± 5 sin t + cos t t. i y = t e (c sin 5 5 t + c cos t). Next we find the particular solution to y + y + y = e t and y + y + y = e t and add them up. For y + y + y = e t, since is not a root of the characteristic equation, let Y = ae t. + Y + Y = a = e t (a + a + a) = e t a = 6 Y = 6 et. For y + y + y = e t, since is not a root of the characteristic equation, let Y = ae t. + Y + Y = e t (a a + a) = e t a = Y = e t. y(t) = e t (c sin 5 5 t + c cos t) + 6 et e t.. First solve the homogeneous equation. y + y = 0 r + = 0 r = ±i y = c cos t + c sin t. Next we find the particular solution to y + y = t and y + y = e t and add them up. For y + y = t, since 0 is not a root of the characteristic equation, let Y = at + bt + c. + Y = a + (at + bt + c) = at + bt + c + a = t a =, b = 0, c = 8 Y = t 8.

5 For y + y = e t, since is not a root of the characteristic equation, let Y = ae t. + Y = e t (a + a) = e t a = 5 Y = 5 et. y(t) = c cos t + c sin t + t et. y (t) = c sin t + c cos t + t + 5 et. y(0) = 0, y (0) = c = 0, c + 5 = c = 9 0, c = 7 0. y(t) = 9 0 cos t sin t + t et.. First solve the homogeneous equation. y + y = 0 r + = 0 r = ±i y = c cos t + c sin t. Next we find the particular solution to y + y = sin t. Since i is a root of the characteristic equation, let Y = at cos t + bt sin t. + Y = a sin t + b cos t = sin t a =, b = 0 Y = t cos t. y(t) = c cos t + c sin t t cos t. y (t) = c sin t + c cos t cos t + t sin t. y(0) =, y (0) = c =, c = c =, c = 8. y(t) = cos t 8 sin t t cos t. Section.6. First solve the homogeneous equation. y + y = 0 r + = 0 r = ±i y = cos t, y = sin t. The Wronskian determinant of y and y is W = y y y y = cos t cos t + sin t sin t =. 5

6 The general solution to y + y = g(t) = tan t is y (t)g(t) y (t)g(t) y(t) = y (t) + y (t) sin t = cos t + sin t sin t = cos t (sec t cos t) + sin t( cos t + c ) cos t = cos t[ln(sec t + tan t) sin t c ] + sin t( cos t + c ) = cos t ln(sec t + tan t) + c cos t + c sin t. 8. First solve the homogeneous equation. y y + y = 0 (r ) = 0 r = r = y = e t, y = te t. The Wronskian determinant of y and y is W = y y y y = e t e t (t + ) te t e t = e t. The general solution to y y + y = g(t) = e t /( + t ) is y (t)g(t) y (t)g(t) y(t) = y (t) + y (t) = e t t + tet + t + t = et [ ln( + t ) c ] + te t [arctan t + c ] = et ln( + t ) + te t arctan t + c e t + c te t. 0. The homogeneous equation is an Euler equation t y y = 0. y = t n n(n ) = (n )(n + ) = 0 y = t, y = t. The Wronskian determinant of y and y is The general solution to W = y y y y = t ( t ) t =. t y t y = g(t) = t is y(t) = y (t) y (t)g(t) t = t t + t = t ln t + + (c )t c y (t)g(t) + y (t) t = t [ln t + t = t ln t + + c t + c 6t + c ] t [t t + c ] t. 6

7 . It s easy to verify that y (t) = t and y (t) = te t are two independent solutions of t y t(t + )y + (t + )y = 0. The Wronskian determinant of y and y is The general solution to W = y y y y = te t (t + ) te t = t e t. y t + t is y (t)g(t) y(t) = y (t) + y (t) tet t = t + te t t e t y + t + y = g(t) = t t y (t)g(t) t t t e t = t(t c ) + te t ( e t + c ) = t t + c t + c te t. 5. The Wronskian determinant is y + 7 t y + 5 t = t. p(t) = 7 t, g(t) = t, y (t) = t. = exp( p(t)) = t 7. The other linearly independent solution is y (t) = y y(t) = et The general solution is y(t) = y (t) et ( t) e t = t 5. y (t)g(t) y (t)g(t) + y (t) = t + C t + C t 5. (.) 6. y ( + t )y + t y = tet. The Wronskian determinant is p(t) = ( + t ), g(t) = tet, y (t) = + t. = exp( The other linearly independent solution is y (t) = y = ( + t) y(t) p(t)) = te t. te t et = ( + t) ( + t) + t = et. The general solution is y (t)g(t) y (t)g(t) et te t (t + ) te y(t) = y (t) + y (t) = (t + ) + e t t te t te t = (t + )( et c ) + e t (te t + c ) = t et + c (t + ) + c e t. 7

Jim Lambers MAT 285 Spring Semester Practice Exam 2 Solution. y(t) = 5 2 e t 1 2 e 3t.

Jim Lambers MAT 285 Spring Semester Practice Exam 2 Solution. y(t) = 5 2 e t 1 2 e 3t. . Solve the initial value problem which factors into Jim Lambers MAT 85 Spring Semester 06-7 Practice Exam Solution y + 4y + 3y = 0, y(0) =, y (0) =. λ + 4λ + 3 = 0, (λ + )(λ + 3) = 0. Therefore, the roots

More information

MATH 23 Exam 2 Review Solutions

MATH 23 Exam 2 Review Solutions MATH 23 Exam 2 Review Solutions Problem 1. Use the method of reduction of order to find a second solution of the given differential equation x 2 y (x 0.1875)y = 0, x > 0, y 1 (x) = x 1/4 e 2 x Solution

More information

Review for Exam 2. Review for Exam 2.

Review for Exam 2. Review for Exam 2. Review for Exam 2. 5 or 6 problems. No multiple choice questions. No notes, no books, no calculators. Problems similar to homeworks. Exam covers: Regular-singular points (5.5). Euler differential equation

More information

Math 308 Exam I Practice Problems

Math 308 Exam I Practice Problems Math 308 Exam I Practice Problems This review should not be used as your sole source for preparation for the exam. You should also re-work all examples given in lecture and all suggested homework problems..

More information

ODE Homework Solutions of Linear Homogeneous Equations; the Wronskian

ODE Homework Solutions of Linear Homogeneous Equations; the Wronskian ODE Homework 3 3.. Solutions of Linear Homogeneous Equations; the Wronskian 1. Verify that the functions y 1 (t = e t and y (t = te t are solutions of the differential equation y y + y = 0 Do they constitute

More information

Math 21B - Homework Set 8

Math 21B - Homework Set 8 Math B - Homework Set 8 Section 8.:. t cos t dt Let u t, du t dt and v sin t, dv cos t dt Let u t, du dt and v cos t, dv sin t dt t cos t dt u v v du t sin t t sin t dt [ t sin t u v ] v du [ ] t sin t

More information

A First Course in Elementary Differential Equations: Problems and Solutions. Marcel B. Finan Arkansas Tech University c All Rights Reserved

A First Course in Elementary Differential Equations: Problems and Solutions. Marcel B. Finan Arkansas Tech University c All Rights Reserved A First Course in Elementary Differential Equations: Problems and Solutions Marcel B. Finan Arkansas Tech University c All Rights Reserved 1 4 The Method of Variation of Parameters Problem 4.1 Solve y

More information

Calculus IV - HW 3. Due 7/ Give the general solution to the following differential equations: y = c 1 e 5t + c 2 e 5t. y = c 1 e 2t + c 2 e 4t.

Calculus IV - HW 3. Due 7/ Give the general solution to the following differential equations: y = c 1 e 5t + c 2 e 5t. y = c 1 e 2t + c 2 e 4t. Calculus IV - HW 3 Due 7/13 Section 3.1 1. Give the general solution to the following differential equations: a y 25y = 0 Solution: The characteristic equation is r 2 25 = r 5r + 5. It follows that the

More information

Math 308 Exam I Practice Problems

Math 308 Exam I Practice Problems Math 308 Exam I Practice Problems This review should not be used as your sole source of preparation for the exam. You should also re-work all examples given in lecture and all suggested homework problems..

More information

dt 2 roots r = 1 and r =,1, thus the solution is a linear combination of e t and e,t. conditions. We havey(0) = c 1 + c 2 =5=4 and dy (0) = c 1 + c

dt 2 roots r = 1 and r =,1, thus the solution is a linear combination of e t and e,t. conditions. We havey(0) = c 1 + c 2 =5=4 and dy (0) = c 1 + c MAE 305 Assignment #3 Solutions Problem 9, Page 8 The characteristic equation for d y,y =0isr, = 0. This has two distinct roots r = and r =,, thus the solution is a linear combination of e t and e,t. That

More information

Homework Solutions: , plus Substitutions

Homework Solutions: , plus Substitutions Homework Solutions: 2.-2.2, plus Substitutions Section 2. I have not included any drawings/direction fields. We can see them using Maple or by hand, so we ll be focusing on getting the analytic solutions

More information

Solutions to Homework 3

Solutions to Homework 3 Solutions to Homework 3 Section 3.4, Repeated Roots; Reduction of Order Q 1). Find the general solution to 2y + y = 0. Answer: The charactertic equation : r 2 2r + 1 = 0, solving it we get r = 1 as a repeated

More information

Polytechnic Institute of NYU MA 2132 Final Practice Answers Fall 2012

Polytechnic Institute of NYU MA 2132 Final Practice Answers Fall 2012 Polytechnic Institute of NYU MA Final Practice Answers Fall Studying from past or sample exams is NOT recommended. If you do, it should be only AFTER you know how to do all of the homework and worksheet

More information

Chapter 13: General Solutions to Homogeneous Linear Differential Equations

Chapter 13: General Solutions to Homogeneous Linear Differential Equations Worked Solutions 1 Chapter 13: General Solutions to Homogeneous Linear Differential Equations 13.2 a. Verifying that {y 1, y 2 } is a fundamental solution set: We have y 1 (x) = cos(2x) y 1 (x) = 2 sin(2x)

More information

Diff. Eq. App.( ) Midterm 1 Solutions

Diff. Eq. App.( ) Midterm 1 Solutions Diff. Eq. App.(110.302) Midterm 1 Solutions Johns Hopkins University February 28, 2011 Problem 1.[3 15 = 45 points] Solve the following differential equations. (Hint: Identify the types of the equations

More information

Higher Order Linear Equations

Higher Order Linear Equations C H A P T E R 4 Higher Order Linear Equations 4.1 1. The differential equation is in standard form. Its coefficients, as well as the function g(t) = t, are continuous everywhere. Hence solutions are valid

More information

Differential Equations: Homework 2

Differential Equations: Homework 2 Differential Equations: Homework Alvin Lin January 08 - May 08 Section.3 Exercise The direction field for provided x 0. dx = 4x y is shown. Verify that the straight lines y = ±x are solution curves, y

More information

Bessel s Equation. MATH 365 Ordinary Differential Equations. J. Robert Buchanan. Fall Department of Mathematics

Bessel s Equation. MATH 365 Ordinary Differential Equations. J. Robert Buchanan. Fall Department of Mathematics Bessel s Equation MATH 365 Ordinary Differential Equations J. Robert Buchanan Department of Mathematics Fall 2018 Background Bessel s equation of order ν has the form where ν is a constant. x 2 y + xy

More information

1. The graph of a function f is given above. Answer the question: a. Find the value(s) of x where f is not differentiable. Ans: x = 4, x = 3, x = 2,

1. The graph of a function f is given above. Answer the question: a. Find the value(s) of x where f is not differentiable. Ans: x = 4, x = 3, x = 2, 1. The graph of a function f is given above. Answer the question: a. Find the value(s) of x where f is not differentiable. x = 4, x = 3, x = 2, x = 1, x = 1, x = 2, x = 3, x = 4, x = 5 b. Find the value(s)

More information

Math 308 Week 8 Solutions

Math 308 Week 8 Solutions Math 38 Week 8 Solutions There is a solution manual to Chapter 4 online: www.pearsoncustom.com/tamu math/. This online solutions manual contains solutions to some of the suggested problems. Here are solutions

More information

Ex. 1. Find the general solution for each of the following differential equations:

Ex. 1. Find the general solution for each of the following differential equations: MATH 261.007 Instr. K. Ciesielski Spring 2010 NAME (print): SAMPLE TEST # 2 Solve the following exercises. Show your work. (No credit will be given for an answer with no supporting work shown.) Ex. 1.

More information

Section 5.5 More Integration Formula (The Substitution Method) 2 Lectures. Dr. Abdulla Eid. College of Science. MATHS 101: Calculus I

Section 5.5 More Integration Formula (The Substitution Method) 2 Lectures. Dr. Abdulla Eid. College of Science. MATHS 101: Calculus I Section 5.5 More Integration Formula (The Substitution Method) 2 Lectures College of Science MATHS : Calculus I (University of Bahrain) Integrals / 7 The Substitution Method Idea: To replace a relatively

More information

Second Order Linear Equations

Second Order Linear Equations October 13, 2016 1 Second And Higher Order Linear Equations In first part of this chapter, we consider second order linear ordinary linear equations, i.e., a differential equation of the form L[y] = d

More information

HW2 Solutions. MATH 20D Fall 2013 Prof: Sun Hui TA: Zezhou Zhang (David) October 14, Checklist: Section 2.6: 1, 3, 6, 8, 10, 15, [20, 22]

HW2 Solutions. MATH 20D Fall 2013 Prof: Sun Hui TA: Zezhou Zhang (David) October 14, Checklist: Section 2.6: 1, 3, 6, 8, 10, 15, [20, 22] HW2 Solutions MATH 20D Fall 2013 Prof: Sun Hui TA: Zezhou Zhang (David) October 14, 2013 Checklist: Section 2.6: 1, 3, 6, 8, 10, 15, [20, 22] Section 3.1: 1, 2, 3, 9, 16, 18, 20, 23 Section 3.2: 1, 2,

More information

Elementary Differential Equations, Section 2 Prof. Loftin: Practice Test Problems for Test Find the radius of convergence of the power series

Elementary Differential Equations, Section 2 Prof. Loftin: Practice Test Problems for Test Find the radius of convergence of the power series Elementary Differential Equations, Section 2 Prof. Loftin: Practice Test Problems for Test 2 SOLUTIONS 1. Find the radius of convergence of the power series Show your work. x + x2 2 + x3 3 + x4 4 + + xn

More information

Math 266 Midterm Exam 2

Math 266 Midterm Exam 2 Math 266 Midterm Exam 2 March 2st 26 Name: Ground Rules. Calculator is NOT allowed. 2. Show your work for every problem unless otherwise stated (partial credits are available). 3. You may use one 4-by-6

More information

Mathematics 104 Fall Term 2006 Solutions to Final Exam. sin(ln t) dt = e x sin(x) dx.

Mathematics 104 Fall Term 2006 Solutions to Final Exam. sin(ln t) dt = e x sin(x) dx. Mathematics 14 Fall Term 26 Solutions to Final Exam 1. Evaluate sin(ln t) dt. Solution. We first make the substitution t = e x, for which dt = e x. This gives sin(ln t) dt = e x sin(x). To evaluate the

More information

144 Chapter 3. Second Order Linear Equations

144 Chapter 3. Second Order Linear Equations 144 Chapter 3. Second Order Linear Equations PROBLEMS In each of Problems 1 through 8 find the general solution of the given differential equation. 1. y + 2y 3y = 0 2. y + 3y + 2y = 0 3. 6y y y = 0 4.

More information

0.1 Problems to solve

0.1 Problems to solve 0.1 Problems to solve Homework Set No. NEEP 547 Due September 0, 013 DLH Nonlinear Eqs. reducible to first order: 1. 5pts) Find the general solution to the differential equation: y = [ 1 + y ) ] 3/. 5pts)

More information

Ma 221 Final Exam Solutions 5/14/13

Ma 221 Final Exam Solutions 5/14/13 Ma 221 Final Exam Solutions 5/14/13 1. Solve (a) (8 pts) Solution: The equation is separable. dy dx exy y 1 y0 0 y 1e y dy e x dx y 1e y dy e x dx ye y e y dy e x dx ye y e y e y e x c The last step comes

More information

First Order Differential Equations f ( x,

First Order Differential Equations f ( x, Chapter d dx First Order Differential Equations f ( x, ).1 Linear Equations; Method of Integrating Factors Usuall the general first order linear equations has the form p( t ) g ( t ) (1) where pt () and

More information

Math 250 Skills Assessment Test

Math 250 Skills Assessment Test Math 5 Skills Assessment Test Page Math 5 Skills Assessment Test The purpose of this test is purely diagnostic (before beginning your review, it will be helpful to assess both strengths and weaknesses).

More information

Nonconstant Coefficients

Nonconstant Coefficients Chapter 7 Nonconstant Coefficients We return to second-order linear ODEs, but with nonconstant coefficients. That is, we consider (7.1) y + p(t)y + q(t)y = 0, with not both p(t) and q(t) constant. The

More information

Math 180, Final Exam, Fall 2012 Problem 1 Solution

Math 180, Final Exam, Fall 2012 Problem 1 Solution Math 80, Final Exam, Fall 0 Problem Solution. Find the derivatives of the following functions: (a) ln(ln(x)) (b) x 6 + sin(x) e x (c) tan(x ) + cot(x ) (a) We evaluate the derivative using the Chain Rule.

More information

Problem 1 (Equations with the dependent variable missing) By means of the substitutions. v = dy dt, dv

Problem 1 (Equations with the dependent variable missing) By means of the substitutions. v = dy dt, dv V Problem 1 (Equations with the dependent variable missing) By means of the substitutions v = dy dt, dv dt = d2 y dt 2 solve the following second-order differential equations 1. t 2 d2 y dt + 2tdy 1 =

More information

PRELIM 2 REVIEW QUESTIONS Math 1910 Section 205/209

PRELIM 2 REVIEW QUESTIONS Math 1910 Section 205/209 PRELIM 2 REVIEW QUESTIONS Math 9 Section 25/29 () Calculate the following integrals. (a) (b) x 2 dx SOLUTION: This is just the area under a semicircle of radius, so π/2. sin 2 (x) cos (x) dx SOLUTION:

More information

µ = e R p(t)dt where C is an arbitrary constant. In the presence of an initial value condition

µ = e R p(t)dt where C is an arbitrary constant. In the presence of an initial value condition MATH 3860 REVIEW FOR FINAL EXAM The final exam will be comprehensive. It will cover materials from the following sections: 1.1-1.3; 2.1-2.2;2.4-2.6;3.1-3.7; 4.1-4.3;6.1-6.6; 7.1; 7.4-7.6; 7.8. The following

More information

17.2 Nonhomogeneous Linear Equations. 27 September 2007

17.2 Nonhomogeneous Linear Equations. 27 September 2007 17.2 Nonhomogeneous Linear Equations 27 September 2007 Nonhomogeneous Linear Equations The differential equation to be studied is of the form ay (x) + by (x) + cy(x) = G(x) (1) where a 0, b, c are given

More information

Chapter 4: Higher Order Linear Equations

Chapter 4: Higher Order Linear Equations Chapter 4: Higher Order Linear Equations MATH 351 California State University, Northridge April 7, 2014 MATH 351 (Differential Equations) Ch 4 April 7, 2014 1 / 11 Sec. 4.1: General Theory of nth Order

More information

Tangent Lines Sec. 2.1, 2.7, & 2.8 (continued)

Tangent Lines Sec. 2.1, 2.7, & 2.8 (continued) Tangent Lines Sec. 2.1, 2.7, & 2.8 (continued) Prove this Result How Can a Derivative Not Exist? Remember that the derivative at a point (or slope of a tangent line) is a LIMIT, so it doesn t exist whenever

More information

Solutions of Math 53 Midterm Exam I

Solutions of Math 53 Midterm Exam I Solutions of Math 53 Midterm Exam I Problem 1: (1) [8 points] Draw a direction field for the given differential equation y 0 = t + y. (2) [8 points] Based on the direction field, determine the behavior

More information

Series Solutions Near a Regular Singular Point

Series Solutions Near a Regular Singular Point Series Solutions Near a Regular Singular Point MATH 365 Ordinary Differential Equations J. Robert Buchanan Department of Mathematics Fall 2018 Background We will find a power series solution to the equation:

More information

APPLIED MATHEMATICS. Part 1: Ordinary Differential Equations. Wu-ting Tsai

APPLIED MATHEMATICS. Part 1: Ordinary Differential Equations. Wu-ting Tsai APPLIED MATHEMATICS Part 1: Ordinary Differential Equations Contents 1 First Order Differential Equations 3 1.1 Basic Concepts and Ideas................... 4 1.2 Separable Differential Equations................

More information

Review For the Final: Problem 1 Find the general solutions of the following DEs. a) x 2 y xy y 2 = 0 solution: = 0 : homogeneous equation.

Review For the Final: Problem 1 Find the general solutions of the following DEs. a) x 2 y xy y 2 = 0 solution: = 0 : homogeneous equation. Review For the Final: Problem 1 Find the general solutions of the following DEs. a) x 2 y xy y 2 = 0 solution: y y x y2 = 0 : homogeneous equation. x2 v = y dy, y = vx, and x v + x dv dx = v + v2. dx =

More information

Nonhomogeneous Linear Equations: Variation of Parameters Professor David Levermore 17 October 2004 We now return to the discussion of the general case

Nonhomogeneous Linear Equations: Variation of Parameters Professor David Levermore 17 October 2004 We now return to the discussion of the general case Nonhomogeneous Linear Equations: Variation of Parameters Professor David Levermore 17 October 2004 We now return to the discussion of the general case L(t)y = a 0 (t)y + a 1 (t)y + a 2 (t)y = b(t). (1.1)

More information

A: Brief Review of Ordinary Differential Equations

A: Brief Review of Ordinary Differential Equations A: Brief Review of Ordinary Differential Equations Because of Principle # 1 mentioned in the Opening Remarks section, you should review your notes from your ordinary differential equations (odes) course

More information

Find the Fourier series of the odd-periodic extension of the function f (x) = 1 for x ( 1, 0). Solution: The Fourier series is.

Find the Fourier series of the odd-periodic extension of the function f (x) = 1 for x ( 1, 0). Solution: The Fourier series is. Review for Final Exam. Monday /09, :45-:45pm in CC-403. Exam is cumulative, -4 problems. 5 grading attempts per problem. Problems similar to homeworks. Integration and LT tables provided. No notes, no

More information

3. Identify and find the general solution of each of the following first order differential equations.

3. Identify and find the general solution of each of the following first order differential equations. Final Exam MATH 33, Sample Questions. Fall 7. y = Cx 3 3 is the general solution of a differential equation. Find the equation. Answer: y = 3y + 9 xy. y = C x + C x is the general solution of a differential

More information

A = (a + 1) 2 = a 2 + 2a + 1

A = (a + 1) 2 = a 2 + 2a + 1 A = (a + 1) 2 = a 2 + 2a + 1 1 A = ( (a + b) + 1 ) 2 = (a + b) 2 + 2(a + b) + 1 = a 2 + 2ab + b 2 + 2a + 2b + 1 A = ( (a + b) + 1 ) 2 = (a + b) 2 + 2(a + b) + 1 = a 2 + 2ab + b 2 + 2a + 2b + 1 3 A = (

More information

1 Arithmetic calculations (calculator is not allowed)

1 Arithmetic calculations (calculator is not allowed) 1 ARITHMETIC CALCULATIONS (CALCULATOR IS NOT ALLOWED) 1 Arithmetic calculations (calculator is not allowed) 1.1 Check the result Problem 1.1. Problem 1.2. Problem 1.3. Problem 1.4. 78 5 6 + 24 3 4 99 1

More information

Math 250B Final Exam Review Session Spring 2015 SOLUTIONS

Math 250B Final Exam Review Session Spring 2015 SOLUTIONS Math 5B Final Exam Review Session Spring 5 SOLUTIONS Problem Solve x x + y + 54te 3t and y x + 4y + 9e 3t λ SOLUTION: We have det(a λi) if and only if if and 4 λ only if λ 3λ This means that the eigenvalues

More information

Test #2 Math 2250 Summer 2003

Test #2 Math 2250 Summer 2003 Test #2 Math 225 Summer 23 Name: Score: There are six problems on the front and back of the pages. Each subpart is worth 5 points. Show all of your work where appropriate for full credit. ) Show the following

More information

A.P. Calculus BC Test Two Section One Multiple-Choice Calculators Allowed Time 40 minutes Number of Questions 15

A.P. Calculus BC Test Two Section One Multiple-Choice Calculators Allowed Time 40 minutes Number of Questions 15 A.P. Calculus BC Test Two Section One Multiple-Choice Calculators Allowed Time 40 minutes Number of Questions 15 The scoring for this section is determined by the formula [C (0.25 I)] 1.8 where C is the

More information

Form A. 1. Which of the following is a second-order, linear, homogenous differential equation? 2

Form A. 1. Which of the following is a second-order, linear, homogenous differential equation? 2 Form A Math 4 Common Part of Final Exam December 6, 996 INSTRUCTIONS: Please enter your NAME, ID NUMBER, FORM designation, and INDEX NUMBER on your op scan sheet. The index number should be written in

More information

Math 334 A1 Homework 3 (Due Nov. 5 5pm)

Math 334 A1 Homework 3 (Due Nov. 5 5pm) Math 334 A1 Homework 3 Due Nov. 5 5pm No Advanced or Challenge problems will appear in homeworks. Basic Problems Problem 1. 4.1 11 Verify that the given functions are solutions of the differential equation,

More information

MATH 3321 Sample Questions for Exam 3. 3y y, C = Perform the indicated operations, if possible: (a) AC (b) AB (c) B + AC (d) CBA

MATH 3321 Sample Questions for Exam 3. 3y y, C = Perform the indicated operations, if possible: (a) AC (b) AB (c) B + AC (d) CBA MATH 33 Sample Questions for Exam 3. Find x and y so that x 4 3 5x 3y + y = 5 5. x = 3/7, y = 49/7. Let A = 3 4, B = 3 5, C = 3 Perform the indicated operations, if possible: a AC b AB c B + AC d CBA AB

More information

ODE Homework Series Solutions Near an Ordinary Point, Part I 1. Seek power series solution of the equation. n(n 1)a n x n 2 = n=0

ODE Homework Series Solutions Near an Ordinary Point, Part I 1. Seek power series solution of the equation. n(n 1)a n x n 2 = n=0 ODE Homework 6 5.2. Series Solutions Near an Ordinary Point, Part I 1. Seek power series solution of the equation y + k 2 x 2 y = 0, k a constant about the the point x 0 = 0. Find the recurrence relation;

More information

Homework Problem Answers

Homework Problem Answers Homework Problem Answers Integration by Parts. (x + ln(x + x. 5x tan 9x 5 ln sec 9x 9 8 (. 55 π π + 6 ln 4. 9 ln 9 (ln 6 8 8 5. (6 + 56 0/ 6. 6 x sin x +6cos x. ( + x e x 8. 4/e 9. 5 x [sin(ln x cos(ln

More information

Integration by Substitution

Integration by Substitution November 22, 2013 Introduction 7x 2 cos(3x 3 )dx =? 2xe x2 +5 dx =? Chain rule The chain rule: d dx (f (g(x))) = f (g(x)) g (x). Use the chain rule to find f (x) and then write the corresponding anti-differentiation

More information

Differential Equations Practice: 2nd Order Linear: Nonhomogeneous Equations: Undetermined Coefficients Page 1

Differential Equations Practice: 2nd Order Linear: Nonhomogeneous Equations: Undetermined Coefficients Page 1 Differential Equations Practice: 2nd Order Linear: Nonhomogeneous Equations: Undetermined Coefficients Page 1 Questions Example (3.5.3) Find a general solution of the differential equation y 2y 3y = 3te

More information

Math 4B Notes. Written by Victoria Kala SH 6432u Office Hours: T 12:45 1:45pm Last updated 7/24/2016

Math 4B Notes. Written by Victoria Kala SH 6432u Office Hours: T 12:45 1:45pm Last updated 7/24/2016 Math 4B Notes Written by Victoria Kala vtkala@math.ucsb.edu SH 6432u Office Hours: T 2:45 :45pm Last updated 7/24/206 Classification of Differential Equations The order of a differential equation is the

More information

Differential Equations Practice: 2nd Order Linear: Nonhomogeneous Equations: Variation of Parameters Page 1

Differential Equations Practice: 2nd Order Linear: Nonhomogeneous Equations: Variation of Parameters Page 1 Differential Equations Practice: 2nd Order Linear: Nonhomogeneous Equations: Variation of Parameters Page Questions Example (3.6.) Find a particular solution of the differential equation y 5y + 6y = 2e

More information

Second In-Class Exam Solutions Math 246, Professor David Levermore Thursday, 31 March 2011

Second In-Class Exam Solutions Math 246, Professor David Levermore Thursday, 31 March 2011 Second In-Class Exam Solutions Math 246, Professor David Levermore Thursday, 31 March 211 (1) [6] Give the interval of definition for the solution of the initial-value problem d 4 y dt 4 + 7 1 t 2 dy dt

More information

Solution to Review Problems for Midterm II

Solution to Review Problems for Midterm II Solution to Review Problems for Midterm II Midterm II: Monday, October 18 in class Topics: 31-3 (except 34) 1 Use te definition of derivative f f(x+) f(x) (x) lim 0 to find te derivative of te functions

More information

GUIDELINES FOR THE METHOD OF UNDETERMINED COEFFICIENTS

GUIDELINES FOR THE METHOD OF UNDETERMINED COEFFICIENTS GUIDELINES FOR THE METHOD OF UNDETERMINED COEFFICIENTS Given a constant coefficient linear differential equation a + by + cy = g(t), where g is an exponential, a simple sinusoidal function, a polynomial,

More information

Review session Midterm 1

Review session Midterm 1 AS.110.109: Calculus II (Eng) Review session Midterm 1 Yi Wang, Johns Hopkins University Fall 2018 7.1: Integration by parts Basic integration method: u-sub, integration table Integration By Parts formula

More information

Math 152 Take Home Test 1

Math 152 Take Home Test 1 Math 5 Take Home Test Due Monday 5 th October (5 points) The following test will be done at home in order to ensure that it is a fair and representative reflection of your own ability in mathematics I

More information

Assignment # 7, Math 370, Fall 2018 SOLUTIONS: y = e x. y = e 3x + 4xe 3x. y = e x cosx.

Assignment # 7, Math 370, Fall 2018 SOLUTIONS: y = e x. y = e 3x + 4xe 3x. y = e x cosx. Assignment # 7, Math 370, Fall 2018 SOLUTIONS: Problem 1: Solve the equations (a) y 8y + 7y = 0, (i) y(0) = 1, y (0) = 1. Characteristic equation: α 2 8α+7 = 0, = 64 28 = 36 so α 1,2 = (8 ±6)/2 and α 1

More information

Section 4.8 Anti Derivative and Indefinite Integrals 2 Lectures. Dr. Abdulla Eid. College of Science. MATHS 101: Calculus I

Section 4.8 Anti Derivative and Indefinite Integrals 2 Lectures. Dr. Abdulla Eid. College of Science. MATHS 101: Calculus I Section 4.8 Anti Derivative and Indefinite Integrals 2 Lectures College of Science MATHS 101: Calculus I (University of Bahrain) 1 / 28 Indefinite Integral Given a function f, if F is a function such that

More information

MA 162 FINAL EXAM PRACTICE PROBLEMS Spring Find the angle between the vectors v = 2i + 2j + k and w = 2i + 2j k. C.

MA 162 FINAL EXAM PRACTICE PROBLEMS Spring Find the angle between the vectors v = 2i + 2j + k and w = 2i + 2j k. C. MA 6 FINAL EXAM PRACTICE PROBLEMS Spring. Find the angle between the vectors v = i + j + k and w = i + j k. cos 8 cos 5 cos D. cos 7 E. cos. Find a such that u = i j + ak and v = i + j + k are perpendicular.

More information

4r 2 12r + 9 = 0. r = 24 ± y = e 3x. y = xe 3x. r 2 6r + 25 = 0. y(0) = c 1 = 3 y (0) = 3c 1 + 4c 2 = c 2 = 1

4r 2 12r + 9 = 0. r = 24 ± y = e 3x. y = xe 3x. r 2 6r + 25 = 0. y(0) = c 1 = 3 y (0) = 3c 1 + 4c 2 = c 2 = 1 Mathematics MATB44, Assignment 2 Solutions to Selected Problems Question. Solve 4y 2y + 9y = 0 Soln: The characteristic equation is The solutions are (repeated root) So the solutions are and Question 2

More information

Math Exam 2, October 14, 2008

Math Exam 2, October 14, 2008 Math 96 - Exam 2, October 4, 28 Name: Problem (5 points Find all solutions to the following system of linear equations, check your work: x + x 2 x 3 2x 2 2x 3 2 x x 2 + x 3 2 Solution Let s perform Gaussian

More information

A Brief Review of Elementary Ordinary Differential Equations

A Brief Review of Elementary Ordinary Differential Equations A A Brief Review of Elementary Ordinary Differential Equations At various points in the material we will be covering, we will need to recall and use material normally covered in an elementary course on

More information

and verify that it satisfies the differential equation:

and verify that it satisfies the differential equation: MOTIVATION: Chapter One: Basic and Review Why study differential equations? Suppose we know how a certain quantity changes with time (for example, the temperature of coffee in a cup, the number of people

More information

California State University Northridge MATH 280: Applied Differential Equations Midterm Exam 2

California State University Northridge MATH 280: Applied Differential Equations Midterm Exam 2 California State University Northridge MATH 280: Applied Differential Equations Midterm Exam 2 November 3, 203. Duration: 75 Minutes. Instructor: Jing Li Student Name: Student number: Take your time to

More information

MA 266 Review Topics - Exam # 2 (updated)

MA 266 Review Topics - Exam # 2 (updated) MA 66 Reiew Topics - Exam # updated Spring First Order Differential Equations Separable, st Order Linear, Homogeneous, Exact Second Order Linear Homogeneous with Equations Constant Coefficients The differential

More information

LECTURE 14: REGULAR SINGULAR POINTS, EULER EQUATIONS

LECTURE 14: REGULAR SINGULAR POINTS, EULER EQUATIONS LECTURE 14: REGULAR SINGULAR POINTS, EULER EQUATIONS 1. Regular Singular Points During the past few lectures, we have been focusing on second order linear ODEs of the form y + p(x)y + q(x)y = g(x). Particularly,

More information

Math 23 Practice Quiz 2018 Spring

Math 23 Practice Quiz 2018 Spring 1. Write a few examples of (a) a homogeneous linear differential equation (b) a non-homogeneous linear differential equation (c) a linear and a non-linear differential equation. 2. Calculate f (t). Your

More information

M343 Homework 6. Enrique Areyan May 31, 2013

M343 Homework 6. Enrique Areyan May 31, 2013 M343 Homework 6 Enrique Areyan May 31, 013 Section 3.5. + y + 5y = 3sin(t). The general solution is given by: y h : Characteristic equation: r + r + 5 = 0 r = 1 ± i. The solution in this case is: y h =

More information

Mar 10, Calculus with Algebra and Trigonometry II Lecture 14Undoing the Marproduct 10, 2015 rule: integration 1 / 18

Mar 10, Calculus with Algebra and Trigonometry II Lecture 14Undoing the Marproduct 10, 2015 rule: integration 1 / 18 Calculus with Algebra and Trigonometry II Lecture 14 Undoing the product rule: integration by parts Mar 10, 2015 Calculus with Algebra and Trigonometry II Lecture 14Undoing the Marproduct 10, 2015 rule:

More information

This practice exam is intended to help you prepare for the final exam for MTH 142 Calculus II.

This practice exam is intended to help you prepare for the final exam for MTH 142 Calculus II. MTH 142 Practice Exam Chapters 9-11 Calculus II With Analytic Geometry Fall 2011 - University of Rhode Island This practice exam is intended to help you prepare for the final exam for MTH 142 Calculus

More information

Math 201 Solutions to Assignment 1. 2ydy = x 2 dx. y = C 1 3 x3

Math 201 Solutions to Assignment 1. 2ydy = x 2 dx. y = C 1 3 x3 Math 201 Solutions to Assignment 1 1. Solve the initial value problem: x 2 dx + 2y = 0, y(0) = 2. x 2 dx + 2y = 0, y(0) = 2 2y = x 2 dx y 2 = 1 3 x3 + C y = C 1 3 x3 Notice that y is not defined for some

More information

3. Identify and find the general solution of each of the following first order differential equations.

3. Identify and find the general solution of each of the following first order differential equations. Final Exam MATH 33, Sample Questions. Fall 6. y = Cx 3 3 is the general solution of a differential equation. Find the equation. Answer: y = 3y + 9 xy. y = C x + C is the general solution of a differential

More information

Mathematics for Chemistry: Exam Set 1

Mathematics for Chemistry: Exam Set 1 Mathematics for Chemistry: Exam Set 1 June 18, 017 1 mark Questions 1. The minimum value of the rank of any 5 3 matrix is 0 1 3. The trace of an identity n n matrix is equal to 1-1 0 n 3. A square matrix

More information

Non-homogeneous equations (Sect. 3.6).

Non-homogeneous equations (Sect. 3.6). Non-homogeneous equations (Sect. 3.6). We study: y + p(t) y + q(t) y = f (t). Method of variation of parameters. Using the method in an example. The proof of the variation of parameter method. Using the

More information

Series Solution of Linear Ordinary Differential Equations

Series Solution of Linear Ordinary Differential Equations Series Solution of Linear Ordinary Differential Equations Department of Mathematics IIT Guwahati Aim: To study methods for determining series expansions for solutions to linear ODE with variable coefficients.

More information

First Order Differential Equations

First Order Differential Equations Chapter First Order Differential Equations Contents. The Method of Quadrature.......... 68. Separable Equations............... 74.3 Linear Equations I................ 85.4 Linear Equations II...............

More information

Math 2Z03 - Tutorial # 6. Oct. 26th, 27th, 28th, 2015

Math 2Z03 - Tutorial # 6. Oct. 26th, 27th, 28th, 2015 Math 2Z03 - Tutorial # 6 Oct. 26th, 27th, 28th, 2015 Tutorial Info: Tutorial Website: http://ms.mcmaster.ca/ dedieula/2z03.html Office Hours: Mondays 3pm - 5pm (in the Math Help Centre) Tutorial #6: 3.4

More information

Section 4.7: Variable-Coefficient Equations

Section 4.7: Variable-Coefficient Equations Cauchy-Euler Equations Section 4.7: Variable-Coefficient Equations Before concluding our study of second-order linear DE s, let us summarize what we ve done. In Sections 4.2 and 4.3 we showed how to find

More information

VANDERBILT UNIVERSITY. MATH 2610 ORDINARY DIFFERENTIAL EQUATIONS Practice for test 1 solutions

VANDERBILT UNIVERSITY. MATH 2610 ORDINARY DIFFERENTIAL EQUATIONS Practice for test 1 solutions VANDERBILT UNIVERSITY MATH 2610 ORDINARY DIFFERENTIAL EQUATIONS Practice for test 1 solutions The first test will cover all material discussed up to (including) section 4.5. Important: The solutions below

More information

dx n a 1(x) dy

dx n a 1(x) dy HIGHER ORDER DIFFERENTIAL EQUATIONS Theory of linear equations Initial-value and boundary-value problem nth-order initial value problem is Solve: a n (x) dn y dx n + a n 1(x) dn 1 y dx n 1 +... + a 1(x)

More information

كلية العلوم قسم الرياضيات المعادالت التفاضلية العادية

كلية العلوم قسم الرياضيات المعادالت التفاضلية العادية الجامعة اإلسالمية كلية العلوم غزة قسم الرياضيات المعادالت التفاضلية العادية Elementary differential equations and boundary value problems المحاضرون أ.د. رائد صالحة د. فاتن أبو شوقة 1 3 4 5 6 بسم هللا

More information

More Techniques. for Solving First Order ODE'S. and. a Classification Scheme for Techniques

More Techniques. for Solving First Order ODE'S. and. a Classification Scheme for Techniques A SERIES OF CLASS NOTES FOR 2005-2006 TO INTRODUCE LINEAR AND NONLINEAR PROBLEMS TO ENGINEERS, SCIENTISTS, AND APPLIED MATHEMATICIANS DE CLASS NOTES 1 A COLLECTION OF HANDOUTS ON FIRST ORDER ORDINARY DIFFERENTIAL

More information

Graded and supplementary homework, Math 2584, Section 4, Fall 2017

Graded and supplementary homework, Math 2584, Section 4, Fall 2017 Graded and supplementary homework, Math 2584, Section 4, Fall 2017 (AB 1) (a) Is y = cos(2x) a solution to the differential equation d2 y + 4y = 0? dx2 (b) Is y = e 2x a solution to the differential equation

More information

Equations with regular-singular points (Sect. 5.5).

Equations with regular-singular points (Sect. 5.5). Equations with regular-singular points (Sect. 5.5). Equations with regular-singular points. s: Equations with regular-singular points. Method to find solutions. : Method to find solutions. Recall: The

More information

Nonhomogeneous Equations and Variation of Parameters

Nonhomogeneous Equations and Variation of Parameters Nonhomogeneous Equations Variation of Parameters June 17, 2016 1 Nonhomogeneous Equations 1.1 Review of First Order Equations If we look at a first order homogeneous constant coefficient ordinary differential

More information

MATH 307: Problem Set #7

MATH 307: Problem Set #7 MATH 307: Problem Set #7 Due on: Feb 11, 2016 Problem 1 First-order Variation of Parameters The method of variation of parameters uses the homogeneous solutions of a linear ordinary differential equation

More information

Study 5.5, # 1 5, 9, 13 27, 35, 39, 49 59, 63, 69, 71, 81. Class Notes: Prof. G. Battaly, Westchester Community College, NY Homework.

Study 5.5, # 1 5, 9, 13 27, 35, 39, 49 59, 63, 69, 71, 81. Class Notes: Prof. G. Battaly, Westchester Community College, NY Homework. Goals: 1. Recognize an integrand that is the derivative of a composite function. 2. Generalize the Basic Integration Rules to include composite functions. 3. Use substitution to simplify the process of

More information

Infinite series, improper integrals, and Taylor series

Infinite series, improper integrals, and Taylor series Chapter Infinite series, improper integrals, and Taylor series. Determine which of the following sequences converge or diverge (a) {e n } (b) {2 n } (c) {ne 2n } (d) { 2 n } (e) {n } (f) {ln(n)} 2.2 Which

More information

20D - Homework Assignment 4

20D - Homework Assignment 4 Brian Bowers (TA for Hui Sun) MATH 0D Homework Assignment November, 03 0D - Homework Assignment First, I will give a brief overview of how to use variation of parameters. () Ensure that the differential

More information